On the variational formulation of systems with non-holonomic constraints

TL;DR

This paper extends a variational formulation approach to systems with non-holonomic constraints by using an adapted frame and covariant derivatives, demonstrated through a rolling disc example.

Contribution

It introduces a novel variational formulation for non-holonomic systems using anholonomic frames and covariant derivatives, expanding previous methods for non-conservative systems.

Findings

Successfully formulated non-holonomic systems variationally

Applied method to a rolling disc without slipping

Showed the effectiveness of anholonomic frames in variational principles

Abstract

In a previous article by the author, it was shown that one could effectively give a variational formulation to non-conservative mechanical systems by starting with the first variation functional instead of an action functional. In this article, it is shown that this same approach will also allow one to give a variational formulation to systems with non-holonomic constraints. The key is to use an adapted anholonomic local frame field in the formulation, which then implies the replacement of ordinary derivatives with covariant ones. The method is then applied to the case of a vertical disc rolling without slipping or friction on a plane.